## Characteristics and the three gap theorem.(English)Zbl 0709.11011

Let $$[x]$$ and $$\{x\}$$ denote the integer and fractional parts of the positive real number $$x$$. If $$d_m=[(m+1)x] - [mx]$$ then $$[mx]=\sum^{m-1}_{i=1}d_i+[x]$$. Since each $$d_i$$ is $$[x]$$ or $$[x]+1$$, we can define the characteristic of $$x$$ to be the string $$x_1x_2x_3\cdots$$ where $$x_i=s$$ if $$d_i=[x]$$ and $$x_i=\ell$$ if $$d_i=[x]+1$$ $$(s$$ for small and $$\ell$$ for large). The authors exhibit a connection, for some $$x$$, between the characteristic of $$x$$ and the sequence of arcs or gaps formed by partitioning a circle by successive placements of points by an angle of $$x$$ revolutions. They do this by making use of the three gap theorem, which asserts that points placed in this way partition a circle into gaps of 2 or 3 different lengths.

### MSC:

 11K06 General theory of distribution modulo $$1$$ 11A55 Continued fractions 11B83 Special sequences and polynomials

### Keywords:

greatest integer; fractional parts; three gap theorem
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